Time to Approach Similarity

Webber, Joseph J.; Huppert, Herbert E.

doi:10.1093/qjmam/hbz019

Cited by 4 publications

(5 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…With these two intuitive observations in mind, we can guess that p decreases with τ and diverges as p → 0 + . The exact analytic dependence is not straightforward, but Ball & Huppert (2019) and Webber & Huppert (2019) showed that the inversely proportional relationship (τ ∝ p −1 ) works as an excellent first-order approximation for the radially symmetric case. Although the exact proportionality for the power-law channel might differ from the radially symmetric case, we can assume that…”

Section: Outflow From the Originmentioning

confidence: 99%

“…The method of similarity calculates the behaviour of the current using scaling symmetry, and the solution derived represents the current in a stationary condition when the memory of the initial conditions are lost, which within a finite time scale, differs from the real flow depending on how it is released. Ball & Huppert (2019) and Webber & Huppert (2019) discuss the time scale at which an axisymmetric current converges to the self-similar solution. A similar construction is used herein to investigate the time scale of real flow approaching the similarity solutions, and we found different leading-power terms from the axisymmetric case studied by Ball & Huppert (2019) and Webber & Huppert (2019).…”

Section: Introductionmentioning

confidence: 99%

“…Ball & Huppert (2019) and Webber & Huppert (2019) discuss the time scale at which an axisymmetric current converges to the self-similar solution. A similar construction is used herein to investigate the time scale of real flow approaching the similarity solutions, and we found different leading-power terms from the axisymmetric case studied by Ball & Huppert (2019) and Webber & Huppert (2019).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Shallow current of viscous fluid flowing between diverging or converging walls

Liu,

Huppert

2023

J. Fluid Mech.

View full text Add to dashboard Cite

We investigate the shallow flow of viscous fluid into and out of a channel whose gap width increases as a power law ( $x^n$ ), where $x$ is the downstream axis. The fluid flows slowly, while injected at a rate in the form of $t^\alpha$ , where $t$ is time and $\alpha$ is a constant. The invading fluid has a higher viscosity than the ambient fluid, thus avoiding Saffman–Taylor instability. Similarity solutions of the first kind for the outflow problem are found using approximations of lubrication theory. Zheng et al. (J. Fluid Mech., vol. 747, 2014, pp. 218–246) studied the deep-channel case and found divergent behaviour of the similarity variable as $n\rightarrow 1$ and $n\rightarrow 3$ , when fluid flows into and out of the channel, respectively. No divergence is found in the shallow case presented here up to the breakdown of the geometric assumption. The characteristic equilibration time for the numerically simulated constant-volume flow to converge to the similarity solution is calculated assuming an inverse dependence on the ratio disagreement between the current front using the method of lines. An inverse power dependence between equilibration time and ratio disagreement is found for channels of different powers. A similarity solution of the second kind for the inflow problem is found using the phase-plane formalism and the bisection method. An exponential decay relationship is found between $n$ and the degree $\delta$ of the similarity variable $xt^{-\delta }$ , which does not show any divergent behaviour for large $n$ . An asymptotic behaviour is found for $\delta$ that approaches $1/2$ for $n\gg 1$ .

show abstract

Section: Outflow From the Originmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Shallow current of viscous fluid flowing between diverging or converging walls

Liu,

Huppert

2023

J. Fluid Mech.

View full text Add to dashboard Cite

show abstract

“…The time taken to approach this one-eighth power law is evidently very short in this experiment. This 'adjustment time' has been recently investigated by Ball & Huppert (2019) and Webber & Huppert (2020) for the case of gravitational spread of a viscous fluid (Huppert 1982). They found that this time, somewhat independent of initial conditions, can be just fractions of a second in laboratory experiments, but many days in geological situations, where the extremely viscous lava domes from volcanic eruptions can eventually spread horizontally to hundreds of kilometres.…”

Section: The Squeezing Problem (F > 0)mentioning

confidence: 99%

Spreading or contraction of viscous drops between plates: single, multiple or annular drops

Moffatt¹,

Guest²,

Huppert

2021

J. Fluid Mech.

View full text Add to dashboard Cite

The behaviour of a viscous drop squeezed between two horizontal planes (a squeezed Hele-Shaw cell) is treated by both theory and experiment. When the squeezing force $F$ is constant and surface tension is neglected, the theory predicts ultimate growth of the radius $a\sim t^{1/8}$ with time $t$. This theory is first reviewed and found to be in excellent agreement with experiment. Surface tension at the drop boundary reduces the interior pressure, and this effect is included in the analysis, although it is negligibly small in the squeezing experiments. An initially elliptic drop tends to become circular as $t$ increases. More generally, the circular evolution is found to be stable under small perturbations. If, on the other hand, the force is reversed ($F<0$), so that the plates are drawn apart (the ‘contraction’, or ‘lifting plate’, problem), the boundary of the drop is subject to a fingering instability on a scale determined by surface tension. The effect of a trapped air bubble at the centre of the drop is then considered. The annular evolution of the drop under constant squeezing is still found to follow a ‘one-eighth’ power law, but this is unstable, the instability originating at the boundary of the air bubble, i.e. the inner boundary of the annulus. The air bubble is realised experimentally in two ways: first by simply starting with the drop in the form of an annulus, as nearly circular as possible; and second by forcing four initially separate drops to expand and merge, a process that involves the resolution of ‘contact singularities’ by surface tension. If the plates are drawn apart, the evolution is still subject to the fingering instability driven from the outer boundary of the annulus. This instability is realised experimentally by levering the plates apart at one corner: fingering develops at the outer boundary and spreads rapidly to the interior as the levering is slowly increased. At a later stage, before ultimate rupture of the film and complete separation of the plates, fingering spreads also from the boundary of any interior trapped air bubble, and small cavitation bubbles appear in the very low-pressure region, far from the point of leverage. This exotic behaviour is discussed in the light of the foregoing theoretical analysis.

show abstract

“…an analytical function [5]. Note that such a solution is not expected to hold close to the initial state, being therefore valid on a relatively long timescale [27,28]. This theoretical solution can be used as long as capillary effects can be disregarded, i.e., the height of the current is larger than the capillary length.…”

Section: Introductionmentioning

confidence: 99%